Simplify the following expression and state the condition under which the simplification is valid. $x = \dfrac{-5n^3 - 10n^2 + 120n}{6n^2 - 12n - 288}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ x = \dfrac {-5n(n^2 + 2n - 24)} {6(n^2 - 2n - 48)} $ $ x = -\dfrac{5n}{6} \cdot \dfrac{n^2 + 2n - 24}{n^2 - 2n - 48} $ Next factor the numerator and denominator. $ x = - \dfrac{5n}{6} \cdot \dfrac{(n + 6)(n - 4)}{(n + 6)(n - 8)}$ Assuming $n \neq -6$ , we can cancel the $n + 6$ $ x = - \dfrac{5n}{6} \cdot \dfrac{n - 4}{n - 8}$ Therefore: $ x = \dfrac{ -5n(n - 4)}{ 6(n - 8)}$, $n \neq -6$